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Actual frame observations realize the nominal channel contractions

Lax342547.RawContractions · concepts/Lax342547/RawContractions.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    The full cross forms of two raw units are their actual plus/minus Gram pairings. Their primal/channel contractions agree with the original trace contraction of frame tensors, on the whole profile space and hence on every effective profile where a table is extended.

    Concept map
    27 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 observation_blocks proven

    2 raw_contraction proven

    3 raw_star_contraction proven

    Lean source view on GitHub

    1import Lax342547.TableContractions
    2
    3/-!
    4---
    5title: Actual frame observations realize the nominal channel contractions
    6type: lemma
    7---
    8The full cross forms of two raw units are their actual plus/minus Gram
    9pairings. Their primal/channel contractions agree with the original trace
    10contraction of frame tensors, on the whole profile space and hence on
    11every effective profile where a table is extended.
    12-/
    13
    14namespace Lax342547.RawContractions
    15
    16open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ExactPins Lax342547.ProjectedPins
    17open Lax342547.TableSpaces Lax342547.SmallTables Lax342547.TableContractions
    18open Lax342547.ReferencePins
    19
    20variable {Comp B H N : Type} [Fintype B] [Fintype H] [Fintype N] {E : Matrix B B Binary}
    21
    22def rawForms (oA oB : Fin 2 → Comp → Frame B H N E) : CrossForms Comp B H :=
    23 ⟨fun e => (dotProductBilin Binary Binary).compl₁₂ (observation oA (e, true))
    24 (observation oB (e, false)),
    25 fun e => (dotProductBilin Binary Binary).compl₁₂ (observation oB (e, true))
    26 (observation oA (e, false))⟩
    27
    28axiom observation_blocks (o : Fin 2 → Comp → Frame B H N E) (e : Comp) (s : Bool) (i : Fin 2)
    29 (v : B → Binary) (w : H → Binary) :
    30 observation o (e, s) (primalEmbedding i v) =
    31 (if s then (o i e).P.mulVec v else (o i e).Q.mulVec v) ∧
    32 observation o (e, s) (channelEmbedding i w) =
    33 (if s then (o i e).X.mulVec w else (o i e).Y.mulVec w)
    34
    35axiom raw_contraction [Fintype Comp] (oA oB : Fin 2 → Comp → Frame B H N E)
    36 (X : Submodule Binary (Comp → Matrix B B Binary)) (i z : Fin 2) (x : X) :
    37 fullContraction (rawForms oA oB) X i z x =
    38 profileContraction (oB z) (profileMap (oA i) x.val)
    39
    40axiom raw_star_contraction [Fintype Comp] (oA oB : Fin 2 → Comp → Frame B H N E)
    41 {P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N} (T : Table P Q)
    42 (hT : TableContractions.Extends (rawForms oA oB) T)
    43 (X : Submodule Binary (Comp → Matrix B B Binary)) (i z : Fin 2) (x : effective X P i) :
    44 starContraction T X i z x = profileContraction (oB z) (profileMap (oA i) x.val.val)
    45
    46end Lax342547.RawContractions
    47
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